EXISTENCE AND GLOBAL ATTRACTIVITY OF PERIODIC SOLUTIONS IN A HIGHER ORDER DIFFERENCE EQUATION

Consider the following higher order difference equation x(n + 1) = f(n,x(n)) + g(n, x(n - k)), n = 0, 1, ... when f(n, x) and g(n, x) : {0, 1, ...} x vertical bar 0, infinity) -> vertical bar 0, infinity) are continuous functions in x and periodic functions in n with period p, and k is a nonnegative integer. We show the existence of a periodic solution {(x) over tilde (n)} under certain conditions, and then establish a sufficient condition for {(x) over tilde (n)} to be a global attractor of all nonnegative solutions of the equation. Applications to Riccati difference equation and some other difference equations derived from mathematical biology are also given.